The Cardioid: the basis for all tornado outbreaks???

[Lanny Dean]

I do not know that this might belong on a different thread but in association of the cardioid matter, I bring up the mesocyclone. That unique cloud structure has caught my attention from a point of mathematical mimicking what might be called a hyperboloid of one sheet variation. My point concerns the volume of the mesocyclone cloud in a hyperboloid of one sheet nature because that volume is

piabH[1+1/3(H^2/c^2)]

The reason I bring this up is the 1/3piabH because that is the volume of a cone and don't tornadoes assume a shape of the cone? So, one starts with that cloud shape and volume of a mesocyclone and one has only to note that when the value H^2/c^2 goes to unity, one is left with the volume

4piabH/3

which could be interpreted as an updraft composed of right circular cylinder plus cone and note that it doesn't necessarily involve vorticity as a cause but might include it asn an effect. One can then take such a volume of a hyperboloid of one sheet and use it as a differential equation of the nature

y''+y'+piabH[1+1/3(H^2/c^2)]-4piabH/3=0

and solve! Any takers?

Dave,
I am confused now.
You clearly refered to the "open" Cardioid and gave the full, albeit incorrect equation and yet you have just given another equation for the base of vorticity?
Also, I noticed that you have not yet given a response regarding my first post. Do you intend to do so?

As far as the first cusps, in which you were no doubt referring to,
your thought process still does not explain the events I gave originally including the outbreak of 3/27/08.
Simply stated, if your thought process is correct, then the Holly Colorado tornado should have never happened. In laymens terms: this would be due to the cell being on the "wrong side" of the cusps and depending on the placement, completly out of the "circle" How would you explain this?
In looking at the same outbreak, the Bird city tornado (which many of us here were on personally) would also have been "out of phase".

In fact, after looking at some recent outbreaks this afternoon, most outbreaks do not fit your process, at least in the basic sense.

Admittedly, in most cases of very long track tornado families ie. 5/4/07,
5/3/99, 5/4/03 and 5/29/04, there does seem to be a connection with the Cardioid and placement of the next family or tornado. However, some process still can not be explained by the Cardioid, such as the anticyclonic tornado around the parent tornado or the like. The reasons are many but again, in laymens terms: this process would be well after the cusp and already ongoing before the second cusp.

The basic "right hand rule" still applies...it has to in dealing with any angular velocity. Otherwise, as stated before, you would have mass chaos.
I am sorry but I just can not agree with your thought process.
Maybe you could provide more than just an equation to help?

 

[Skip Talbot]

Lanny, you seem to be one of the few who can make any sense of this. Since our original poster will not do so, can you label a reports graphic to demonstrate these cardioid tornado paths?

 

[Lanny Dean]

Lanny, you seem to be one of the few who can make any sense of this. Since our original poster will not do so, can you label a reports graphic to demonstrate these cardioid tornado paths?

Sure Skip. I will certianly try to get something together....give me just a bit.

 

[Wes Carter]

Alright, I am with ya!

Outside of the latitude and longitude plus time data, how could one set up such an analysis that would be satisfying? I had recommended the Palm Sunday outbreak of April 11, 1965. In fact, I would recommend repeating those isochrone profiles as found in the original Bradbury and Fujita, SMRP #86, University of Chicago study, of which the Monthly Weather Review story is based. Given the time plus earth grid coordinates, what would be satisfying?

I'm trying to follow your logic, but those of us who are not math wizards are having a hard time. When you mentioned isochrone profiles (which I admittedly had to Google, I could not remember what isochrone profiles were, which are explained here) I remembered a discussion in the Tennessee Valley Chasers Community forum here on Stormtrack where Randy Bowers mentioned the Hybrid Single Particle Lagrangian Integrated Trajectory Model from NOAA's Air Resources Laboratory. The isochrone profile is also known as the tautochrone curve problem which has three solutions, one of which is the Langranian Solution.

But I cannot resolve how the cardioid pattern fits into a dispersion pattern that NOAA's HSPLIT Model tries to emulate. Please fill in the holes for us.

Maybe it would be better if you wrote a paper and presented it to us with your sources and methods listed.

 

[Bob Hartig]

Bob,
Although I am well versed in the calculus (I just got a math degree and took many years of calculus courses) I myself am confused as to exactly what Mr. van Grun is trying to say. I haven't seen one picture yet. However, if it helps with the background material, check out the Wikipedia article on cardioids to get an idea for what the shape is and what parametric and polar equations can be used to graph them. If you have a graphing calculator like a TI-83 or 89, you can graph them there. They're pretty graphs.

Thanks, Jeff. The Wiki was enlightening, and I at least now know what a cardioid is in this context. I'll leave the equations alone for the moment as my mind just doesn't wrap itself around such stuff very well. But I'll be following this thread with interest, particularly since it sounds as if Lanny is about to provide some helpful visuals.

 

[Lanny Dean]

Lets see if this helps, keep in mind this is in it's most elementary form.
Note: I will have to edit this just a bit so please be patient with me... here goes:

Lets see if this helps, keep in mind this is in it's most elementary form.
Note: I will have to edit this just a bit so please be patient with me... here goes:
The first graphic could represent the 3-28-07 outbreak where A= the first tornado "family" B= the second, C= the thrid and so on until we reach the "end" or G

We understand that the first example IS A TRUE CARDIOID in it's first "phase" again, in laymens terms, then by nature we can assume that the second "phase" of the Cardioid will rotate again and reproduce A, B, C, D, and so on. SEE SECOND GRAPHIC.
Dave's thought process, at least by my understanding, is that you can determine the next tornado family by following the pattern of the cardioid. The "cusps" referred to are very interesting due to the fact that we understand this as being cyclic in our world. (over and over) The second cusp or complete "cycle" could be considered the start of another tornado family (by his definition)
The only problem is he is not taking into account or simply not including the circumference of the whole. Nor has he included as to what direction.
Example: If A is the tornado that happened near Sharon Springs KS on this date (3/28/07) then B would be the Edison tornado, C should be a tornado near Colby or Menio hypothetically.
As we all know, this was not the case....the next tornado was the Bird City wedge and then of course, Benkelman Nebraska. Both of which are not only out of the cusps but are also completly out of the next "phase"..... as was the Holly Colorado event.

This makes no sense to me.

I could and have put this pattern to test this afternoon on the most memoriable and fearful tornadoes that I have been on, the Greensburg event. Interestingly enough it did seem to "follow" this pattern but....so did the 5-3-99 event. I say the 5-3-99 event, but actually only two tornadoes of that event followed this pattern and that was storm A and B.
Greensburg has a much better pattern to it and you can actually follow that tornado event right along the curve (FWIW)
However, looking at the cardioid pattern, you cannot in any way follow any of the tornadoes after storm B. THEY ARE OFF THE CUSPS or they are before the next "phase"

Remebering that :
The cardioid is the pedal curve of the circle with respect to a point on its circumference. The pedal is the locus of the foot of the perpendicular from a fixed point P on the tangent to the curve. The circle is its own pedal with respect to its centre (because the radius is always perpendicular to a tangent).

Knowing this, his thought process does not make sense to me.

 

[Lanny Dean]

Skip,
Hope the above post helps a little.....anymore than this is above my head.

 

[Travis Stone]

Lanny,

I don't understand the math, but the concept? If that makes sense. How does this apply to the the Memphis storms. I noticed that the Tornado's were a equal distance apart prior to hitting Memphis.

 

[Dave Van Grun]

I am simply talking about the area enclosed by the family of storms.

 

[Dave Van Grun]

I am impressed with the visual stimuli which has been displayed on this thread. Maybe this is the right place after all. I had been trying to subject this information and concept to a verification process, hoping that a mathematics department might be interested and I might be able to go forward, but nobody was interested in doing so. You see, I live in a state which isn't interested in such phenomena, not even capriciously, so I have to find both a place that likes series/sequencing mathematics as well as tornadoes similtaneously in order to stimulate such needed interest. Perhaps that meteorologist who conducted me to this website was correct after all!

 

[Dave Van Grun]

As I indicated, I live in a state which isn't up on tornadoes, so I must confess that I am not up on current events, only past outbreaks. But I got a feeling that the Memphis events seem to be likened to what I would call a series/sequence of a -sin which is short for x=-(2sin x-sin2x) and y=(2cosx-cos2x) from a cursory point of view!

 

[Jeff Duda]

I'm sorry for adding to the confusion, but Lanny, that first image you showed makes no sense to me. What is being denoted by those fat black lines? Is it actual tornado tracks? I'm having trouble piecing together what that image means. Are those black lines supposed to be pointing out to the graph itself? I see G and F are pretty close, but the rest fall far short of reaching the graph.

 

[Bob Hartig]

I am simply talking about the area enclosed by the family of storms.

Dave, it would be extremely helpful if you would actually show us what you mean. Lanny has done a great job of trying to provide a visual example. Can you improve upon it for us so we can see exactly what you're driving at? Since even those here who are well-versed in calculus are having a hard time grasping your idea clearly, a graphic or two will be much more helpful at this point than trying to explain in words and formulae.

 

[Dave Van Grun]

Let's see if I can simplify this further. I stated that I resolved the area of tornado families into a series/sequence of the cardioid. This is only for 1/2 the total circumference of the cardioid because meteorological conditions are usually solved in a zonal format and hence seldom come back upon themselves. Therefore, the series only goes to 180pi radians or 1/2 of the cardioid(a lobe of the cardioid). Tornado families are thought to be consistent with the location of critical points on that 1/2 circumference, these being at 0,pi/2,2pi/3, and pi of a given areal involvement. The area could be as small as say 700 square miles as in the May 26, 1963 Oklahoma outbreak or as large as 20,000 square miles as in the Super outbreak of '74!

 

[Jacob Ferden]

While the detail of the math is way over my head at the moment (in Calc I right now), my main question revolves around the possible application of such a theory. Is this something that could actually be applied to forecasting a major outbreak? For instance, could you actually make an advance forecast of something similar to Greensburg? If not, then, while it is a very interesting concept, I fail to see the practical relevance.